13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Figure 2. Scheme of the piezoelectric machine 3. Calorimetric analysis 3.1. Model of heat diffusion A numerical model was built to estimate the evolution of intrinsic dissipation from temperature measurement fields during the fatigue test. We focused on the longitudinal distribution of heat sources within the specimen gauge part. A 1D calorimetric analysis has been justified assuming in a first approximation a uniaxial tension-compression stress state. From the heat equation: CT k T s − ∆ = • ρ (1) where T is the temperature, ρ the mass density; C the heat capacity; k the thermal conduction coefficient. s(x,y,z,t) symbolizes the volume of heat source. Following Boulanger’s hypotheses [8], the 1D diffusion equation for a non-constant cross-section can be written as [9]: C s x t S x S x x x t x x t C k x x t t x t D ρ θ θ ρ τ θ θ ( , ) ( ) ( , ) '( ) ² ( , ) ( ) ( , ) ( , ) 2 1 = ∂ ∂ + ∂ ∂ − + ∂ ∂ (2) with θ = T – T° is the temperature change, T° is the room temperature and D1τ is a time constant term which characterizes the heat losses through lateral surfaces of the specimen: ( )) 2 ( ( ) ( ) 1 h e l x C S x x D + ⋅ = ρ τ (3) where e is the specimen thickness, l(x) is its width with respect to x. We note by ( ) ( ) S x e l x = ⋅ the cross-section at this point. The mean dissipation over several thousand cycles was solely estimated
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